/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 30 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))

Q is empty.

----------------------------------------

(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:sieve_1 > zprimes > s_1 > cons_1 > nats_1 > 0 > filter_3

and weight map:

   0=3
   zprimes=8
   cons_1=1
   s_1=1
   sieve_1=0
   nats_1=2
   filter_3=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))




----------------------------------------

(2)
Obligation:
Q restricted rewrite system:
R is empty.
Q is empty.

----------------------------------------

(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
----------------------------------------

(4)
YES